Find the real part of \(\frac{{z + w}}{{z - w}}\).

Find the value of the real part of \(\frac{{z + w}}{{z - w}}\) when \(\left| z \right| = \left| w \right|\).

\(\frac{{z + w}}{{z - w}} = \frac{{\left( {a + c} \right) + {\text{i}}\left( {b + d} \right)}}{{\left( {a - c} \right) + {\text{i}}\left( {b - d} \right)}}\)

\( = \frac{{\left( {a + c} \right) + {\text{i}}\left( {b + d} \right)}}{{\left( {a - c} \right) + {\text{i}}\left( {b - d} \right)}} \times \frac{{\left( {a - c} \right) - {\text{i}}\left( {b - d} \right)}}{{\left( {a - c} \right) - {\text{i}}\left( {b - d} \right)}}\)     M1A1

real part \( = \frac{{\left( {a + c} \right)\left( {a - c} \right) + \left( {b + d} \right)\left( {b - d} \right)}}{{{{\left( {a - c} \right)}^2} + {{\left( {b - d} \right)}^2}}} = \left( {\frac{{{a^2} - {c^2} + {b^2} - {d^2}}}{{{{\left( {a - c} \right)}^2} + {{\left( {b - d} \right)}^2}}}} \right)\)     A1A1

Note: Award A1 for numerator, A1 for denominator.

[4 marks]

\(\left| z \right| = \left| w \right| \Rightarrow {a^2} + {b^2} = {c^2} + {d^2}\)     R1

hence real part = 0      A1

Note: Do not award R0A1.

[2 marks]